**The ultimate desire of mankind is to identify wholeness, to grasp the essence of being, to be integrated with the harmony, perfection, patterns, and cycles of the material, metaphorical and metaphysical worlds. This desire motivates us to explore the realms of fact and fancy, logic and metaphor, reason and emotion, to capture the whole of being in one part, to see it, hear it, feel it, and enjoy it in everyday life. The circle is an object of nature, an idealization of pure mathematics, and a symbol or framework we use to understand and describe our world. The circle exists independently of human thought, as ripples in a pond, or the appearance of the sun and moon, or the shape of the iris of an eye. In mathematics, we choose to define a circle as the places at a constant distance from a center, usually in two dimensions. **

The Yin-Yang symbol of two parts spiraling within a circle is a traditional icon of Confucianism and Taoism. It suggests movement around the inside of the circle. It also provides a paradigm of polarity with which to view the dynamics of everyday life. As a symbol, it can be as personal and internal as a heart, which gives and receives blood through each complete cycle. It can also be as general and external as the cycles of day and night.

The **yin-yang** symbol of Confucianism and Taoism

The Buddhist circular mandala designs have been used continuously for millennia. "A mandala (Sanskrit for "circle") is a symbolic diagram of the universe, arranged in circles, used in tantric Buddhism. The Swiss psychologist Carl Jung considered the mandala to be a universally occurring pattern associated with the mythological representation of the self.

The Mahakala Gonpo-Magpo chakra mandala, by A. T. Mann

Zoroastrianism, the tradition of ancient Persia, is believed by scholars to have been influential in the later development of metaphysical concepts in Abrahamic and Eastern religious beliefs. Its influence survives to our own day, not only in central Asia, but in such products of the European post-Romantic movement as Richard Strauss's music and Friedrich Nietzsche's book, "Thus Spoke Zarathustra." Modern historians have dated the time of Zoroaster to approximately 1750 BC. Also known as Zarathustra, he was the founder of the Zoroastrian tradition. One symbol of Zoroastrianism is the Fra-vahar, a figure that stands for the ideal moral and spiritual focus in life. Fra is the direction, forward, and vahar describes a pulling force. Of the two circles in the figure, the ring in the hand is a reminder that we are bound to keep our promises or agreements with others. The other circle, at the waist, reminds us that our spirits live on, in essence immortal, and so also symbolizes infinity.

The **Fra-vahar** symbol of Zoroastrianism

The mystical theologian Cardinal Nicholas of Cusa regarded mathematics as the best symbol for things divine. He says in

**De Docta Ignorantia**:

**Since there is no other approach to a knowledge of things divine than that of symbols, we cannot do better than use mathematical signs on account of their indestructible certitude**

For example, Cusa used geometry to illustrate the identity of the circle and the line. As a circle becomes very large, it appears less curved, much like how the surface of the Earth appears flat to us because it is so large. In the limit where the circle becomes infinite, then the curvature vanishes and the circle coincides with the straight line.

Cusa's correspondence between the circle and the line, however, has two disadvantages. First, it does not include the point at infinity. Second, it requires passing through an infinite process to form the correspondence. Our approach will therefore differ from Cusa's, even though it follows his basic insight that the apparent opposites of the line circle can be identified. As we will see below, there is another mathematical correspondence between the circle and the line which includes the point at infinity and requires no infinite process. The correspondence is essentially a transformation of our point of view so that the line is seen as a circle. This shift in perspective reveals that the line discontinuously separated from the point at infinity is equivalent to a single continuous circle.

We begin by drawing a vertical z-axis through the line to form a Cartesian coordinate system with the origin of the x-axis at (0,0) and the point at infinity at (0,−1). Now draw a circle of radius 1 with its center at the origin. Note that the point at infinity corresponds to the bottom point on the circle. In addition, the points (−1,0) and (1,0) on the line correspond to points on the circle. There is thus a self-evident correspondence between three points on the circle and three points of the linear mandala. Moreover, there is a one-to-one correspondence between all the other points on the line and all the other points on the circle.

To see this correspondence, imagine a line rotating around the pivot point (0,-1) or, if you prefer, an infinite number of lines radiating outward from (0,-1), the point at infinity. Each of these lines intersects the x-axis at a single point and also intersects the circle at a single point. In other words, each line creates a one-to-one correspondence between a point on the line and a point on the circle. This means that the circle is equivalent to the line.

Notice that there is one line that does not actually intersect the x-axis: the horizontal line parallel to the x-axis. This line does, however, intersect the point p at infinity, which is also a unique point on the circle. This line, therefore, matches these two points. Thus, the line plus the point at infinity is equivalent to the entire circle: every point on this circular mandala is matched with one unique point in the linear mandala. Moreover, this correspondence is continuous, meaning that it matches nearby points on the line with nearby points on the circle. In technical terms, this continuous equivalence of the line to the circle is expressed more precisely by saying that the extended real line is homeomorphic to the circle, i.e., they are topologically isomorphic. The essential fact to understand is that the line plus the point at infinity is completely equivalent to the circle, so we are perfectly justified in viewing it as really being a circle.